A027640 Poincaré series [or Poincare series] for ring of modular forms of genus 2.

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 2, 0, 4, 0, 4, 0, 5, 0, 6, 0, 8, 0, 7, 0, 10, 0, 11, 0, 12, 0, 14, 1, 17, 0, 16, 1, 21, 1, 22, 1, 24, 2, 27, 3, 31, 2, 31, 4, 37, 4, 39, 5, 42, 6, 46, 8, 52, 7, 52, 10, 60, 11, 63, 12, 67, 14

Offset

0,11

Comments

a(k) for k>0 is the dimension of the space of Siegel modular forms of genus 2 and weight k (for the full modular group Gamma_2). - Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009

References

B. Runge, On Siegel modular forms I, J. Reine Angew. Math., 436 (1993), 57-85.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

J. Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,1,1,0,0,-1,-1,-1,1,0,0,1,-1,-1,-1,0,0,1,1,1,0,0,0,-1).

Formula

G.f.: (1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)).

Mathematica

Table[SeriesCoefficient[Series[(1+t^(35))/((1-t^4) (1-t^6)(1-t^(10)) (1-t^(12))), {t, 0, 100}], i], {i, 0, 100}] (* Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009 *)

Prog

(PARI) Vec((1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)) + O(x^100)) \\ Colin Barker, Jul 27 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)) )); // G. C. Greubel, Aug 04 2022
(Sage)
def A027640_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)) ).list()
A027640_list(100) # G. C. Greubel, Aug 04 2022

Crossrefs

Cf. A165685 for the corresponding dimension of the space of cusp forms. - Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009

Sequence in context: A128145 A128143 A292561 * A349448 A194666 A325799

Adjacent sequences: A027637 A027638 A027639 * A027641 A027642 A027643

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved